Relations between two parts of the spectrum of a Schrödinger operator and other remarks on the absolute continuity of the spectrum in a typical case

Abstract

We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schrödinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is opened from the other by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum. The second part of this article contains a discussion of spectral properties of a family of Schrödiger operators depending on a real parameter $t$. The results claim that the absolutely continuous spectrum of an operator of this family is essentially supported by the positive half-line for almost every $t$.